OFFSET
10,5
COMMENTS
All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 10..1000
FORMULA
G.f.: (1-Product_{j>0} (1-x^(10*j)+x^(11*j))) / (Product_{j>0} (1-x^j)).
EXAMPLE
a(14) = 2, because 2 partitions of 14 contain (at least) one clique of size 10: [1,1,1,1,1,1,1,1,1,1,2,2], [1,1,1,1,1,1,1,1,1,1,4].
MAPLE
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->`if`(j=10, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=10..60);
MATHEMATICA
max = 60; f = (1 - Product[1 - x^(10j) + x^(11j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 10] (* Jean-François Alcover, Oct 01 2014 *)
Table[Length[Select[IntegerPartitions[n], MemberQ[Length/@Split[#], 10]&]], {n, 10, 60}] (* Harvey P. Dale, Oct 02 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jan 05 2011
STATUS
approved